Infinite-dimensional Estabrook–Wahlquist prolongations for the sine-Gordon equation

Abstract

We are looking for the universal covering algebra for all symmetries of a given partial differential equation (PDE), using the sine-Gordon equation as a typical example for a nonevolution equation. For nonevolution equations, Estabrook–Wahlquist prolongation structures for nonlocal symmetries depend on the choice of a specific subideal of the contact module to define the PDE. For each inequivalent such choice we determine the most general solution of the prolongation equations, as subalgebras of the (infinite-dimensional) algebra of all vector fields over the space of nonlocal variables associated with the PDE, in the style of Vinogradov covering spaces. We show explicitly how previously known prolongation structures, known to lie within the Kac–Moody algebra, A(1)1, are special cases of these general solutions, although we are unable to identify the most general solutions with previously studied algebras. We show the existence of gauge transformations between prolongation structures, viewed as determining connections over the solution space, and use these to relate (otherwise) distinct algebras. Faithful realizations of the universal algebra allow integral representations of the prolongation structure, opening up interesting connections with algebras of Toeplitz operators over Banach spaces, an area that has only begun to be explored.